1. Field of the Invention
The present invention relates to a track groove of a ball bearing for a rectilinear motion and, more particularly, to a shape of a track groove of the type which can make the slide resistance small while retaining rigidity.
2. Description of the Prior Art
The shape of the track groove of such a kind of bearing is mainly divided into a circular arc groove shape and a Gothic arch groove shape.
These two track groove shapes have a merit and a demerit in terms of the performance of the bearing, respectively, so that they are individually used in dependence on the conditions when they used.
Generally, when two elastic members I and II are come into contact with each other, they have two sets of curvatures [=1/radius] (.rho.I.sub.1, .rho.I.sub.2) and (.rho.II.sub.1, .rho.II.sub.2) which are perpendicular to each other at contact point. However, according to the Hertz's theory, the shape of the compressive contact surface is an ellipse and when it is assumes that the length of the major axis is 2a and the length of the minor axis is 2b, the values of a and b will be expressed by the following equations. ##EQU1## where, P is a compressive load and .theta.I and .theta.II are elastic constants of the elastic members I and II.
An arrangement of a bearing shown in FIG. 1 is substituted for equations (I) and (II). ##EQU2##
Also, when a Poisson's ratio 1/m assumes 0.3 and a modulus of direct elasticity E assumes 2.12.times.10.sup.6 kg/cm.sup.2, EQU .theta.I=.theta.II=4.times.(m.sup.2 -1)/m.sup.2 .times.E=1.72.times.10.sup.6
Therefore, equations (I) and (II) will become ##EQU3##
The values of .mu. and .nu. can be obtained from the value of the auxiliary variable [=cos .tau.=1/(4.times.f-1)] by way of the conversion table. .mu. is proportional to the auxiliary variable and .nu. is inversely proportional to the auxiliary variable.
From the above equations, there is the relationship among a, b and f such that both a and b increase with an increase in f.
When considering the situation of the elastic contact between the ball and the track groove on the basis of such a fundamental elastic theory, the shapes of conventional respective track grooves have a drawback such that, as shown in FIG. 2 (showing an example of a circular arc groove) and FIG. 3 (showing an example of a Gothic arch groove), there is a large difference between d.sub.1 ad d.sub.2 and the different slip of the ball or the like occurs due to the difference in peripheral velocity (v.sub.1 =(1/2)d.sub.1 w, v.sub.2 =(1/2)d.sub.2 w) between d.sub.1 and d.sub.2. In general, this phenomenon does not cause an increase in slide resistance such as to produce a serious problem because a lubricating material is interposed under many use conditions. However, such an increase amount is disadvantageous under the particularly severe conditions that are required for the drive portions of measuring instruments, bearings of the detecting portions, and the like where it is necessary to suppress the slide resistance to as small a value as possible.